TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-06

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The totality set is Pi_2^0-complete

Statement

$T O T = {e : (\forall x) φ_{e} (x) ↓}$ is $Π_{2}^{0}$ -complete.

Facts & Assumptions

Given: an acceptable numbering and a $Π_{2}^{0}$ set $A$ .

Proof

technique · direct

1.1

The condition $e \in T O T$ is $\forall x \exists s T (e, x, s)$ , hence is $Π_{2}^{0}$ .

givenconstruct

2.1

Write $z \in A ⟺ \forall x \exists y R (z, x, y)$ with $R$ primitive recursive. Uniformly produce a program which on input $x$ searches for such a $y$ and halts when one appears. By s-m-n its index is a total computable function of $z$ , and that program is total exactly when $z \in A$ .

step 1.1construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ludovic Patey, Computability Theory, Example 1.5 (standard reference, not scraped)